Remove giblets from chicken cavity; reserve for another use. Loosen skin over breast of chicken by pushing fingers between skin and meat, taking care not to tear skin. Do not loosen skin over wings and drumsticks. Using rubber spatula or small spoon, spread pesto under breast skin; massage skin to evenly spread pesto. Combine remaining 2 tablespoons oil and lemon juice in small bowl; brush over chicken skin. Insert meat thermometer into center of thickest part of thigh, not touching bone. Tuck wings under back; tie legs together with wet kitchen string.

Place chicken, breast side up, on grid directly over drip pan. Grill, on covered grill, over medium-low coals 1 hour 10 minutes to 1 hour 30 minutes or until thermometer registers 185°F, adding 4 to 9 briquets to both sides of the fire after 45 minutes to maintain medium-low coals. Transfer chicken to carving board; tent with foil. Let stand 15 minutes before carving. Garnish with basil and currants.